Inequality proof (Hilbert space)
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Show that if H is a Hilbert space, then: $$Vert(x+y)Vert^2 - Vert(x - y)Vert^2 le 4 Vert xVert Vert yVert, $$ for all $x, y in H. $
inequality hilbert-spaces
asked Nov 24 at 17:02
Loreen
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Show that if H is a Hilbert space, then: $$Vert(x+y)Vert^2 - Vert(x - y)Vert^2 le 4 Vert xVert Vert yVert, $$ for all $x, y in H. $
inequality hilbert-spaces
asked Nov 24 at 17:02
Loreen
32
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up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Show that if H is a Hilbert space, then: $$Vert(x+y)Vert^2 - Vert(x - y)Vert^2 le 4 Vert xVert Vert yVert, $$ for all $x, y in H. $
inequality hilbert-spaces
asked Nov 24 at 17:02
Loreen
32
Show that if H is a Hilbert space, then: $$Vert(x+y)Vert^2 - Vert(x - y)Vert^2 le 4 Vert xVert Vert yVert, $$ for all $x, y in H. $
inequality hilbert-spaces
inequality hilbert-spaces
asked Nov 24 at 17:02
Loreen
32
asked Nov 24 at 17:02
Loreen
32
asked Nov 24 at 17:02
Loreen
32
asked Nov 24 at 17:02
Loreen
32
asked Nov 24 at 17:02
Loreen
32
32
add a comment |
add a comment |
1 Answer
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Observe that
$$ Vert xpm yVert^2 = langle xpm y,xpm y rangle = Vert x Vert^2 pm 2Re langle x, y rangle + Vert yVert^2, $$
so that
$$ Vert x + yVert^2 - Vert x - yVert^2 = 4 Relangle x, y rangle. $$
Now apply Cauchy-Schwarz.
answered Nov 24 at 17:06
MisterRiemann
5,5941623
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Observe that
$$ Vert xpm yVert^2 = langle xpm y,xpm y rangle = Vert x Vert^2 pm 2Re langle x, y rangle + Vert yVert^2, $$
so that
$$ Vert x + yVert^2 - Vert x - yVert^2 = 4 Relangle x, y rangle. $$
Now apply Cauchy-Schwarz.
answered Nov 24 at 17:06
MisterRiemann
5,5941623
add a comment |
up vote
0
down vote
accepted
Observe that
$$ Vert xpm yVert^2 = langle xpm y,xpm y rangle = Vert x Vert^2 pm 2Re langle x, y rangle + Vert yVert^2, $$
so that
$$ Vert x + yVert^2 - Vert x - yVert^2 = 4 Relangle x, y rangle. $$
Now apply Cauchy-Schwarz.
answered Nov 24 at 17:06
MisterRiemann
5,5941623
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Observe that
$$ Vert xpm yVert^2 = langle xpm y,xpm y rangle = Vert x Vert^2 pm 2Re langle x, y rangle + Vert yVert^2, $$
so that
$$ Vert x + yVert^2 - Vert x - yVert^2 = 4 Relangle x, y rangle. $$
Now apply Cauchy-Schwarz.
answered Nov 24 at 17:06
MisterRiemann
5,5941623
Observe that
$$ Vert xpm yVert^2 = langle xpm y,xpm y rangle = Vert x Vert^2 pm 2Re langle x, y rangle + Vert yVert^2, $$
so that
$$ Vert x + yVert^2 - Vert x - yVert^2 = 4 Relangle x, y rangle. $$
Now apply Cauchy-Schwarz.
answered Nov 24 at 17:06
MisterRiemann
5,5941623
answered Nov 24 at 17:06
MisterRiemann
5,5941623
answered Nov 24 at 17:06
MisterRiemann
5,5941623
answered Nov 24 at 17:06
MisterRiemann
5,5941623
5,5941623
add a comment |
add a comment |
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